Regarding the ruling-out/realization of the loss of compactness alternative in the global theory, as was studied in [1]}, [2]}, the established monotonicity property is strong enough to assert a “compactness or front” result stating that this possibility must manifest as a broadening phenomenon, leading to a monotone front type of solution at the end of the bifurcation curve. When the underlying system possesses a Hamiltonian structure, a so-called conjugate flow analysis can be casted utilizing the conserved quantities to rule out the broadening alternative [3]}, [1]}, [2]}, [6]}, [7]}, [8]}. Moreover, for some particular problems such a Hamiltonian structure may also allow one to obtain uniform bounds on solutions that can account for the realization of broadening [7]}. In the cases we consider, however, the system is not Hamiltonian, and we do not have any obvious conserved quantities that can be of much use to control the solutions. Taking advantage of the monotonicity and together with delicate algebra we are able to prove the nonexistence of monotone front solutions, cf. Lemma REF  and Lemma REF . Using this idea we can also prevent the blowup of solutions \((u, \eta )\)  in the case of bifurcation from stationary waves, which leads to a sharp result ensuring the loss of ellipticity as the only remaining alternative cf. Theorem REF . For the other case of solutions bifurcating from the classical Boussinesq waves, we are able to winnow the alternatives down to the possibilities of either the loss of ellipticity or that the curve continues up to the appearance of an “extreme wave” that has a stagnation point, cf. Theorem REF .
